Rogue Waves of the Kundu-DNLS Equation

doi:10.4236/ojapps.2013.31B1020

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Open Journal of Applied Sciences, 2013, 3, 99-101

doi:10.4236/ojapps.2013.31B1020 Published Online April 2013 (http://www.scirp.org/journal/ojapps)

Rogue Waves of the Kundu-DNLS Equation

Shibao Shan, Chuanzhong Li, Jingsong He

Department of Mathematics, Ningbo University, Ningbo, China

Email: [email protected]

Received 2013

ABSTRACT

In this paper, we give the Lax pair and construct the Darboux transformation of the Kundu-DNLS equation. Further-

more, the rogue wave solutions of the Kundu-DNLS equation are derived by using the Taylor expansion of the breather

solution. What's more, the triangular and the circular patterns of the third rouge solution are displayed.

Keywords: Kundu-DNLS Equation; Darboux Transformation; Rogue Waves

1. Introduction

As one of the most important integrable systems in many

branches of physics and applied mathematics, the deriva-

tive nonlinear Schr\"{o}dinger (DNLS) equation





2 x

xxt qqiqiq



(1)

has been studied in optics, water wave and so on [1,2].

Considering the significance of the higher order nonlin-

earities in physical system, the DNLS equation yields an

integrable higher nonlinear equation, i.e. Kundu-DNLS

equation [3,4]

0)2(

)()(

2*2





QQiQ

QiQQiQiQ

xxxtxxxt





(2)

by means of a nonlinear transformation of the field

with arbitrary gauge function



qeQq 





For example, setting



xdxQ 







, Kundu-DNLS

equation implies Eckhaus-Kundu (EK) equation.

Here denotes the complex conjugate of Q, and α s

a real parameter.

Rogue waves are one of those fascinating destructive

phenomena in nature that have not been fully explained

so far. Understanding the initial conditions that foster

rogue waves could be useful both in attempts to avoid

them by seafarers and in generating highly energetic

pulses in optical fibers. There are several method to

solve the integrable equations, for instance, Hirota

method [5], inverse scattering transformation [6], bilin-

ear method [7], Darboux transformation [8]. In this paper,

we have given the rogue waves of the DNLS and the

coupled system of Hirota and Maxwell-Bloch equations

[2,9,10]. Inspired by the importance of these recent in-

teresting developments about the analysis of rogue waves

of the NLS-type equations, we shall construct the rogue

wave solutions of the Kundu-DNLS equation with the

help of the Darboux transformation.

2. Darboux Transformation and Lax Pair

The Darboux transformation is a powerful method used

to generate the soliton solutions for integrable equations.

Inspired by classical Darboux transformation for the DNLS

equation, we consider the coupled Kundu-DNLS equa-

tion,

0)2(

)()(





RQiQ

QiRQiQiQ

xxxtxxxt





(3)

0)2(

)()(





QRiR

RiQRiRiR

xxxtxxxt





(4)

where



is a arbitrary gauge function. This form of the

equation is very extensive, which is reduced to the equa-

tion (2) for with the sign of the nonlinear term

changed. The Kundu-DNLS equation can be obtained if

α is a real parameter.

QR 

We first present a general framework for deriving the

required conservation rule for the DNLS equation. We

start with the linear set of Lax equations:

,, 







VU tx (5)

where U and V depend on the complex constant eigen-

value parameter λ.









































































Re0

iQRiV

iU i









with





.2)(2







ieQQiQeeQiQeG 

S. B. SHAN ET AL.

100

where λ is the eigenvalue, Φ is the eigenfunction corre-

sponding to λ.

In general, considering the universality of Darboux

transformation, according to the Kundu-DNLS equation

(3) and (4) , we can start from

22









































dc



(6)

where are functions

222211110000,,,,,,,,,,, dcbadcbadcba

From



1TUTUTx (7)

and



1TVTVTt



(8)

then the now solutions









11,RQ are given by



1211

2221



























(9)

Here

222

111

222

111











So far, we discussed about the determinant construc-

tion of one-fold Darboux transformation of Kundu-

DNLS equation. As an application of these transforma-

tions of Kundu-DNLS equation, rogue wave solutions

will be constructed in the next section.

3. Rogue Wave Solutions

In this section, we construct the rogue wave solution of

Kundu-DNLS equation . This kind of solution only ap-

pears in some special region of time and space and then

drown into a fixed non-vanishing plane. By making use

of the Taylor expansion for the breather solution, one

order rogue wave solution of for the Kundu-DNLS

equation is obtained



)2(

txi





 (10)

,8812484

4888883

2334

422222

tititixix

txtxtixtitxxv









.44488

888481

44322

222

tix

txxititxititv





The picture of one order rogue wave solution of the

Kundu-DNLS equation and its corresponding density

graph are plotted in Figure 1.

When we take ,1,1,1,2 











ca the picture of

second rogue wave solution for the Kundu-DNLS equa-

tion is displayed in Figure 2.

Next, we examine third-order rogue waves. In this

case, form the figures, We can get third-order rogue

wave solution with the help of

,1,1,2







ca the

picture of third-order rogue wave solution is displayed in

Figure 3.

Figure 1. The One order rogue wave solution of the Kundu-

DNLS equation with a = –2, c = 1, ξ = 1, η = 1.

Figure 2. The second order rogue wave solution of the Kundu-

DNLS equation with a = –2, c = 1, ξ = 1, η = 0.8.

Figure 3. The third order rogue wave solution |Qr3|2 f the

Kundu-DNLS equation with a = –2, c = 1, ξ = 0.8, η = 0.8.

S. B. SHAN ET AL.

101

(a)

(b)

Figure 4. The third order rogue wave solution |Qr3|2 of the

Kundu-DNLS equation w ith (a) a = –2, c = 1, S0 = 0, S1 =

500, S2 = 0; (b) a = –2, c = 1, S0 = 0, S1 = 0, S2 = 1000.

We can split the third order rogue wave solution into

triangle structure. A particular structure is displayed in

Figure 4(a). The third-order rogue wave is seen to pos-

sess a regular triangle spatial symmetry structure.

What's more, we also can split the third order rogue

wave solution into pentagon structure. A particular struc-

ture is displayed in Figure 4(b). The third-order rogue

wave exhibits a regular pentagon spatial symmetry

structure.

4. Conclusions

In this paper, we construct the Darboux transformation

for the Kundu-DNLS equation. This Darboux transfor-

mation, in particular, allows us to calculate higher order

rogue wave solutions in a unified way. In this way, we

can derive the higher order rogue wave solutions for

Kundu-DNLS equation by making use of the Darboux

transformation. Particularly, these rogue wave solutions

possess several free parameters. With the help of these

parameters, these rogue waves constitute some patterns,

such as fundamental pattern, triangular pattern, circular

pattern.

5. Acknowledgements

This work is supported by the NSF of China under Grant

No.11271210 and K. C. Wong Magna Fund in Ningbo

University. Jingsong He is also supported by Natural

Science Foundation of Ningbo under Grant No.

2011A610179. Chuanzhong Li is supported by the Na-

tional Natural Science Foundation of China under Grant

No.11201251, the Natural Science Foundation of Zheji-

ang Province under Grant No. LY12A01007. We thank

Prof. Yishen Li (USTC, Hefei, China) for his long time

support and useful suggestions.

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